mixing - a primary difficulty with MCMC algorithms

The primary difficulty with MCMC algorithms, however, is the issue of mixing — that is,ensuring that the algorithm does not get 'stuck' in local maxima. Various solutions have been developed to deal with this problem. One of the simplest involves running several copies of the MCMC algorithm in parallel and starting from different points, with pairs of copies switching states from time-to-time⁵⁶. Allowing copies to swap places occasionally means that the parameter space can be explored more efficiently. Other schemes involve augmenting the 'state–space' of the process: we add another variable to the space of parameters in such a way that it is easier for the algorithm to accept new states. For example, a useful idea is to add a 'temperature' to the process. In practice, this might involve mixing a 'hot' chain, which takes more frequent jumps, and a 'cool' chain, in which jumps are rarer. The addition of temperature allows the process to explore the parameter space with less risk of getting stuck; however, this greater efficiency occurs at the cost of the requirement for a more complicated algorithm. In some settings, a single process is run; in others, multiple parallel chains are used^48, 57. Owing to the additional complexity involved, these schemes have yet to be widely embraced within the genetics community.